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Random matrix-improved estimation of covariance matrix distances

Abstract : Given two sets x (1) 1 ,. .. , x (1) n 1 and x (2) 1 ,. .. , x (2) n 2 ∈ R p (or C p) of random vectors with zero mean and positive definite covariance matrices C 1 and C 2 ∈ R p×p (or C p×p), respectively, this article provides novel estimators for a wide range of distances between C 1 and C 2 (along with divergences between some zero mean and covariance C 1 or C 2 probability measures) of the form 1 p n i=1 f (λ i (C −1 1 C 2)) (with λ i (X) the eigenvalues of matrix X). These estimators are derived using recent advances in the field of random matrix theory and are asymptotically consistent as n 1 , n 2 , p → ∞ with non trivial ratios p/n 1 < 1 and p/n 2 < 1 (the case p/n 2 > 1 is also discussed). A first "generic" estimator, valid for a large set of f functions, is provided under the form of a complex integral. Then, for a selected set of atomic functions f which can be linearly combined into elaborate distances of practical interest (namely, f (t) = t, f (t) = ln(t), f (t) = ln(1 + st) and f (t) = ln 2 (t)), a closed-form expression is provided. Beside theoretical findings, simulation results suggest an outstanding performance advantage for the proposed estimators when compared to the classical "plug-in" estimator 1 p n i=1 f (λ i (Ĉ −1 1Ĉ 2)) (withĈ a = 1 n a n a i=1 x (a) i x (a) i), and this even for very small values of n 1 , n 2 , p. A concrete application to kernel spectral clustering of covariance classes supports this claim.
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Romain Couillet, Malik Tiomoko, Steeve Zozor, Eric Moisan. Random matrix-improved estimation of covariance matrix distances. Journal of Multivariate Analysis, Elsevier, 2019, 174, pp.104531. ⟨10.1016/j.jmva.2019.06.009⟩. ⟨hal-02355223⟩

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